Simplify the following expression: $y = \dfrac{3x^2- 4x- 15}{3x + 5}$
Explanation: First use factoring by grouping to factor the expression in the numerator. This expression is in the form ${A}x^2 + {B}x + {C}$ First, find two values, $a$ and $b$ , so: $ \begin{eqnarray} {ab} &=& {A}{C} \\ {a} + {b} &=& {B} \end{eqnarray} $ In this case: $ \begin{eqnarray} {ab} &=& {(3)}{(-15)} &=& -45 \\ {a} + {b} &=& &=& {-4} \end{eqnarray} $ In order to find ${a}$ and ${b}$ , list out the factors of $-45$ and add them together. Remember, since $-45$ is negative, one of the factors must be negative. The factors that add up to ${-4}$ will be your ${a}$ and ${b}$ When ${a}$ is ${5}$ and ${b}$ is ${-9}$ $ \begin{eqnarray} {ab} &=& ({5})({-9}) &=& -45 \\ {a} + {b} &=& {5} + {-9} &=& -4 \end{eqnarray} $ Next, rewrite the expression as $({A}x^2 + {a}x) + ({b}x + {C})$ $ ({3}x^2 +{5}x) + ({-9}x {-15}) $ Factor out the common factors: $ x(3x + 5) - 3(3x + 5)$ Now factor out $(3x + 5)$ $ (3x + 5)(x - 3)$ The original expression can therefore be written: $ \dfrac{(3x + 5)(x - 3)}{3x + 5}$ We are dividing by $3x + 5$ , so $3x + 5 \neq 0$ Therefore, $x \neq -\frac{5}{3}$ This leaves us with $x - 3; x \neq -\frac{5}{3}$.